Distributions of Hook lengths in integer partitions

TL;DR

This paper analyzes the distribution of hook lengths in integer partitions, proving normality for the count of t-hooks and Gamma distribution convergence for multiples of t, with explicit asymptotic mean and variance formulas.

Contribution

It establishes the limiting distributions of hook length counts in partitions, including normal and Gamma distributions, with precise asymptotic parameters.

Findings

Distribution of t-hooks is asymptotically normal with explicit mean and variance.

Number of hook lengths divisible by t converges to a shifted Gamma distribution.

Provides asymptotic formulas for mean and variance of hook length counts.

Abstract

Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number of $t$-hooks in the partitions of $n$. We prove that the limiting distribution is normal with mean $\mu_t(n)\sim \frac{\sqrt{6n}}{\pi}-\frac{t}{2}$ and variance $\sigma_t^2(n)\sim \frac{(\pi^2-6)\sqrt{6n}}{2\pi^3}.$ Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed $t\geq 4$ in partitions of $n$ converge to a shifted Gamma distribution with parameter $k=(t-1)/2$ and scale $\theta=\sqrt{2/(t-1)}.$